Pure and hydrated deep eutectic solvents (DES) are proposed to form self-assembled nanostructures within the fluid bulk, similar to the bicontinuous L3 phase common for ionic liquids (ILs). Labelled choline chloride : urea : water DES were measured using small-angle neutron scattering (SANS), showing no long-range nanostructure. However, solutions of the surfactant AOT in this DES yielded scattering consistent with the L3 "sponge" phase, which was fitted using the Teubner-Strey model. A disclike model gave local structural information, namely, a linear increase in radius versus solvent water content (w = molar ratio of DES : water), eventually forming large, turbid lamellar phases at 10w; an L3-to-Lα transition was observed. Simultaneous multi-contrast SANS fits show the surfactant headgroup region is dominated by interactions with poorly-soluble Na+ at low water contents, and numerically-superior [cholinium]+ as water content increases. The modified interfacial Gaussian curvature from cation : anion volume matching stabilizes the lamellar morphology, allowing the bilayer aggregation number to increase.